A popular way to evaluate algorithms that utilize different
information has emerged from the algorithms community. The idea is to
compute a competitive ratio, which places an on-line
algorithm in competition with an algorithm that receives more
information [674,892]. The idea can generally be
applied to plans. First a cost is formulated, such as the total
distance that the robot travels to solve a navigation task. A
competitive ratio can then be defined as

(12.28)

The maximum is taken over all , which is usually an infinite
set, as in the case of the bug algorithms. A competitive ratio for a
navigation problem can be made by comparing the optimal distance to
the total distance traveled by the robot during the execution of the
on-line algorithm. Since is infinite, many plans fail to produce
a finite competitive ratio. The bug algorithms, while elegant,
represent such an example. Imagine a goal that is very close, but a
large obstacle boundary needs to be explored. An obstacle boundary
can be made arbitrarily large while making the optimal distance to the
goal very small. When evaluated in (12.28), the result
over all environments is unbounded. In some contexts, the ratio may
still be useful if expressed as a function of the representation. For
example, if is a polygon with edges, then an
competitive ratio means that (12.28) is bounded over all
by for some
. For competitive ratio
analysis in the context of bug algorithms, see [375].

Figure 12.26:
(a) A lost cow must find its way to the
gate, but it does not know in which direction the gate lies. (b) If
there is no bound on the distance to the gate, then a doubling spiral
strategy works well, producing a competitive ratio of .

A nice illustration of competitive ratio analysis and issues is
provided by the lost-cow problem [60]. As shown
in Figure 12.26a, a short-sighted cow is following along
an infinite fence and wants to find the gate. This makes a convenient
one-dimensional planning problem. If the location of the gate is
given, then the cow can reach it by traveling directly. If the cow is
told that the gate is exactly distance away, then it can move one
unit in one direction and return to try the other direction if the
gate has not been found. The competitive ratio in this case (the set
of environments corresponds to all gate placements) is . What if
the cow is told only that the gate is at least distance 1 away? In
this case, the best strategy is a spiral search, which is to
zig-zag back and forth while iteratively doubling the distance
traveled in each direction, as shown in Figure 12.26b. In
other words: left one unit, right one unit, left two units, right two
units, left four units, and so on. The competitive ratio for this
strategy turns out to be , which is optimal. This approach
resembles iterative deepening, which was covered in Section
2.2.2.